Models of atmospheric chemistry The concept of model Atmospheric structure and dynamics Chemical processes in the atmosphere Model equations and numerical approaches Formulation of radiative, chemical , and aerosol rates Numerical methods for chemical systems Numerical methods for advection Parameterization of subgrid-scale processes Surface fluxes Atmospheric observations and model evaluation Inverse methods for atmospheric chemistry Appendix E: Brief mathematical review Available at http://acmg.seas.harvard.edu/education Password is ‘ctm’

Models solve the continuity equation for atmospheric species Eulerian flux form: Eulerian advective form: Lagrangian form: Brasseur and Jacob, 4.2

Eulerian models partition atmospheric domain into gridboxes This discretizes the continuity equation in space Solve continuity equation for individual gridboxes Present computational limit -107 gridboxes In global models, this implies a horizontal resolution of ~ 1o (~100 km) in horizontal and ~ 1 km in vertical Chemical Transport Models (CTMs) use external meteorological data as input General Circulation Models (GCMs) compute their own meteorological fields Brasseur and Jacob, 4.8

Lagrangian models track transport of points in model domain (no grid!) Transport large number of points with trajectories from input meteorological data base (U) over time steps Dt Points have mixing ratio or mass but no volume Determine local concentrations in a given volume by the statistics of points within that volume or by interpolation position to+Dt PROS over Eulerian models: stable for any wind speed no error from spatial averaging easily track air parcel histories easy to parallelize CONS: need very large # points for statistics inhomogeneous representation of domain individual trajectories do not mix nonlinear chemistry is problematic UDt position to Brasseur and Jacob, 4.11

Lagrangian receptor-oriented modeling Run Lagrangian model backward from receptor location, with points released at receptor location only flow backward in time Efficient quantification of source influence distribution on receptor (“footprint”)

Question The Eulerian form of the continuity equation is a first-order PDE in four dimensions. What are suitable boundary conditions for each of these dimensions in a global model?

Atmospheric turbulence Reynolds number Re = UL/ >> 104 down to mm scale Turbulence occurs on all scales down to mm where molecular diffusion takes over Typical observations of horizontal wind in PBL Brasseur and Jacob, 8.2

Summer daytime observations at Harvard Forest, MA vertical wind w T CO2 Brasseur and Jacob, 10.2.4

Turbulent kinetic energy cascade wavenumber “Big whirls have little whirls, Which feed on their velocity. And little whirls have lesser whirls And so on to viscosity” Lewis Fry Richardson

Turbulent diffusion parameterization California fire plumes, Oct 2004 Industrial plumes Brasseur and Jacob, 8.4.1

Deep convection Organized but subgrid (except in cloud-resolving model with ~1 km resolution) Requires non-local parameterization Convective cloud (0.1-100 km) detrainment Model vertical levels downdraft Large-scale subsidence updraft entrainment Model grid scale Brasseur and Jacob, 8.7

Questions 1. Deep convection is almost always initiated from the boundary layer (lowest ~2 km of atmosphere), never from the free troposphere above. Why? 2. Aircraft vertical profiles downwind of deep convective events often show a "C-shaped" profile for surface air pollutants with high concentrations in the boundary layer and upper troposphere, and low concentrations in the middle troposphere. Why? 3. Could such a shape be simulated with an eddy diffusion parameterization for deep covenction?